Theory question 110
Triangles and Quadrilaterals(a) Prove that the sum of the angles of a triangle is two right angles. (b) In a triangle LMN, the side MN is produced to P and the bisector of ∠LNP meets ML produced at Q. If ∠LMN = 46° and ∠MLN = 80°, calculate ∠LQN, stating clearly your reasons.
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(a) Given △ABC. Produce BC to X and draw CY parallel to BA.
∠ACY = ∠BAC (alternate angles); ∠YCX = ∠ABC (corresponding angles).
∠ACB + ∠ACY + ∠YCX = 180° (angles on a straight line), so ∠C + ∠A + ∠B = 180° = two right angles.
(b) ∠LNM = 180° − 46° − 80° = 54° (angle sum of a triangle).
∠LNP = 46° + 80° = 126° (exterior angle = sum of interior opposite angles).
∠LNQ = 126° ÷ 2 = 63° (NQ bisects ∠LNP).
In △MNQ: ∠MNQ = 54° + 63° = 117°, so ∠LQN = ∠MQN = 180° − 46° − 117° = 17°.
Find the exterior angle at N first, then halve it.